HYPOTHESIS AND THEORY ARTICLEpublished: 25 February 2014

doi: 10.3389/fphy.2014.00007

Programmable ferroelectric tunnel memristorAndy Quindeau , Dietrich Hesse and Marin Alexe*

Max Planck Institute of Microstructure Physics, Halle (Saale), Germany

Edited by:

Nicholas X. Fang, MassachusettsInstitute of Technology, USA

Reviewed by:

Byungchan Han, Daegu GyeongbukInstitute of Science and Technology,South KoreaPavel Lukashev, University ofNebraska - Lincoln, USA

*Correspondence:

Marin Alexe, Max Planck Institute ofMicrostructure Physics, Weinberg 2,D-06120 Halle (Saale), Germanye-mail: malexe@mpi-halle.mpg.de

We report a programmable analog memristor based on genuine electronic resistiveswitching combining ferroelectric switching and electron tunneling. The tunnel currentthrough an 8 unit cell thick epitaxial Pb(Zr0.2Ti0.8)O3 film sandwiched betweenLa0.7Sr0.3MnO3 and cobalt electrodes obeys the Kolmogorov-Avrami-Ishibashi model forbidimensional growth with a characteristic switching time in the order of 10−7 s. Theanalytical description of switching kinetics allows us to develop a characteristic transferfunction that has only one parameter viz. the characteristic switching time and fullypredicts the resistive states of this type of memristor.

Keywords: memristor, KAI, ferroelectric switching, ferroelectric tunnel junction, multiferroic oxides

INTRODUCTIONIn analogy to the resistor, inductor, and capacitor the mem-ristor (memory resistor) is the passive circuit element, whichwas until recently considered to be “the missing circuit ele-ment” [1, 2]. Memristance, the memristor’s main characteristic,which has the dimension of a resistance, should be tunable andhistory-dependent. Almost 40 years after the theoretical descrip-tion, memristive behavior was firstly demonstrated in nanoscalesystems possessing ionic transport properties [3]. Very recentlygenuine electronic effects, such as the direct tunnel effect in ultra-thin ferroelectric films, so-called ferroelectric tunnel junctions(FTJ), [4] have been proposed as memristors [5, 6]. It has beenshown that the resistance of a ferroelectric tunnel junction isindeed tunable and history-dependent. Nevertheless, program-ming the resistance of such a device in a designed way remainsa major goal to be achieved.

In this letter, we investigate memristor characteristics andswitching kinetics of a memristor based on a ferroelectrictunnel junction that has been shown to obey proper quan-tum tunneling [7, 8] and not thermionic injection, as it hasbeen shown on ferroelectric semiconductors [9]. We demon-strate that the memristive states are univocally related to thepolarization switching and that every memristor state canbe achieved by design. Based on the analytical descriptiongiven by the Kolmogorov-Avrami-Ishibashi [KAI] model [10, 11]we are able to explain the memristive behavior as being aresult of coexisting, parallel-connected ferroelectric domainsof opposite polarity inside 3600 μm2 capacitor devices. Thissaid, in principle every ferroelectric tunnel junction with abigger capacitor area than the nuclei size of its ferroelectricdomains is intrinsically a memristor, irrespective of the injectionmechanism.

MATERIALS AND METHODSAccording to the definition given by Chua [2] the time-dependent characteristics of a charge-controlled memristor canbe described by:

v (t) = M(q(t)

)i(t) (1)

where v (t) is the output voltage characteristics, M the mem-ristance (with the dimension of a resistance) and the time-dependency given through the charge q (t) and the current i(t).In the very specific situation where M = constant, the equationdescribes the behavior of a simple resistor.

In the present case of a ferroelectric tunnel junction, the elec-tronic tunneling probability through the ferroelectric layer andthus the resistance is controlled by the ferroelectric polarization,which is equivalent to a surface charge at the electrodes [12].Therefore, we can state that a FTJ is a charge-controlled mem-ristor and Equation (1) applies. The charge q(t) can now simplybe replaced by the time-dependent polarization P(t):

v (t) = M (P(t)) i(t) (2)

The memristive state of a FTJ is therefore dependent on the fer-roelectric polarization and its time-dependent evolution is univo-cally related to the time dependence of the polarization and it canin principle be predicted. This is based on the working principleof FTJ, i.e., the electric current through a metal-ferroelectric-metal heterostructure is polarization dependent through the bandalignment at the metal-ferroelectric interface [13, 14]. On theother hand, the time-dependent function P(t) is the result of theferroelectric switching process. The polarization switching kinet-ics are usually described in terms of nucleation and growth offerroelectric domains. The most common theoretical descriptionof ferroelectric switching kinetics is based on the growth of nucleithat are randomly distributed across the ferroelectric capacitor[15] by ferroelectric domain wall propagation [16, 17]. After aninitial nucleation process that takes place in the sub-picosecondrange, ferroelectric domains grow laterally until they coalesce andthe entire polarization will be in the opposite direction [18]. Thetime dependence of the switching polarization P is given by theKAI equation: [10]

P/Ps(t) = P(t) = 1 − e−(t/τ)n(3)

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PHYSICS

Quindeau et al. Programmable ferroelectric tunnel memristor

with P(t) = P(t)/Ps being the normalized switched polarization,where Ps is the saturated value of polarization, τ is the character-istic switching time and n the domain switching dimensionality,which describes the domain wall movement. The exponent ncan have ideally only integer values of 1, 2, and 3 represent-ing one-, bi-, and three-dimensional domain growth during theswitching process. From Equations (2) and (3) the memristancecharacteristics should be given by the following function:

M/Ps(t) ≈ 1−e−(t/τ)n(4)

In other words the memristance would be directly proportional tothe amount of polarization switched by an applied external pulseof time interval t.

We have applied the above concept to ferroelectric tunneljunctions based on epitaxial oxide layers. The active layer of thestructure is an ultrathin lead zirconate titanate Pb(Zr0.2Ti0.8)O3

(PZT) film epitaxially grown on a La0.7Sr0.3MnO3 (LSMO)/SrTiO3 (STO) (100) substrate. A high-resolution TEM imageof part of the structure is shown in Figure 1. Rather large-area capacitor-like active devices (60 × 60 μm2) were defined bythermal evaporation of cobalt through a shadow mask.

RESULTSA typical current-voltage characteristic of a FTJ device is shownin Figure 2. This figure shows for the first time a hysteretic behav-ior similar to the i–v data of a TiO2-based memristor discussedby Strukov et al. [3] The switching voltages, viz. the voltages atwhich the resistance switches from the OFF to the ON state andvice-versa, correspond to the coercive fields of the ferroelectricthin film, [19] while the above dynamic current-voltage charac-teristic (see Figure 2) is not necessarily identical to the memristorcharacteristics. The latter should reveal the memristance depen-dence on time as for example given by Equation (4). Assumingthat the memristor states in the FTJ are univocally defined by thepolarization switching we may presume that switching kinetics ofa FTJ is identical with the switching kinetics of polarization inultrathin ferroelectric films.

FIGURE 1 | High-resolution transmission electron micrograph of the

Co/PZT/LSMO tunnel junction used in the present study. The thicknessof PZT is 8 unit cells (about 3.2 nm).

In order to experimentally study this phenomenon we havedetermined the memristor characteristics of a FTJ by measuringthe resistive states induced by voltage pulses with variable widthand amplitude. As the memristance is a history-dependent quan-tity, it is desirable to set the memristor device into a defined statebefore setting any other intermediate state. In the present case weused a negative voltage pulse with constant amplitude and widthas a reset pulse, which sets the FTJ into the ON state defined bythe lowest achievable tunnel resistance RON. From this referencestate the FTJ can be driven into any intermediate state betweenRON and ROFF using a writing pulse, i.e., a positive voltage pulsewith variable amplitude and width. This can be done repeatedlyuntil the polarization is fully switching into the opposite direc-tion driving the junction into the OFF state defined by the highestachievable ROFF resistance. Figure 3A shows the memristor char-acteristics determined using pulses of constant amplitude andvariable width, following the scheme in Figure 3B. Between theRON and ROFF states there is in principle an infinite number ofmemristance states. The time scale of the writing pulses to accessthese intermediate states is in the sub-μs range, which is compa-rable with the switching time of ferroelectric polarization that hasbeen shown in previous studies [20]. While the absolute values ofthe saturated (RON and ROFF) tunnel resistances might slightlyvary, mostly due to additional conducting paths represented bydefects, [21] the RON/ROFF ratio remains constant irrespective ofthe write amplitude and/or width.

To normalize the measured data, we define RN as:

RN = R−RON

ROFF−RON(5)

where RN takes values between 0 and 1. If the ROFF/RON ratio orthe tunneling electro resistance value is very large, i.e., larger than100, Equation (5) can be simplified to RN = 1

ROFF(R − RON).

In Figure 3C, RN is plotted vs. the applied pulse time for dif-ferent amplitude values of the applied pulses. For higher pulsevoltages the system switches faster into the saturated ROFF state,

FIGURE 2 | Current-voltage characteristic of the Co/PZT/LSMO

structure shown in Figure 1. The switching from RON to ROFF states andvice-versa is correlated to ferroelectric switching [11].

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Quindeau et al. Programmable ferroelectric tunnel memristor

FIGURE 3 | Memristor characteristics of a Co/PZT/LSMO tunnel

junction with PZT thickness of 8 u.c. (A) Tunneling resistance as functionof the writing pulse width. The black data points were measured after eachreset pulse, the gray data points after each writing pulse. (B) The appliedvoltage train of the measuring setup: in this case the amplitude and widthof the reset pulse were −6 V and 200 ms, respectively, the writing pulsehad the same amplitude as the reset pulse. The measurement of the tunnelcurrent is performed with an applied voltage of 0.1 V. (C) NormalizedON/OFF ratios for different amplitudes of the write pulse. The lines are fitswith Equation (9).

which we contribute to the field-dependent domain wall velocity[22], leading to a faster unipolar formation of the domains in theferroelectric tunnel barrier.

We now postulate that a memristive state can be describedby a resistance resulting from a parallel connection of individualresistances given by coexisting ferroelectric domains of oppositepolarity inside one capacitor similar to the case of a multilevel

FIGURE 4 | (A) A schematic cross section of the ferroelectric layer withvisible domains. The red areas inside the sketch visualize the domains withpointing down polarization, marked by green arrows. The bright yellow areais the “unswitched” area and defined by polarization pointing up (bluearrows). The black arrows indicate the growth direction of the ferroelectricdomains with opposite polarity. (B) Simplifying (A) to a model of parallelconnected resistances, where R↑ is the summarized resistance of areaswith polarization up and R↓ with polarization down accordingly.

ferroelectric memory [20] (Figure 4). In other words, the effec-tive resistance R of the tunnel junction after applying a certainpulse is given by the equivalent resistance of a circuit of resistorsconnected in parallel, which are defined by the area of domainswith polarization up and down, R↑ and R↓ respectively. The areaswhich define R↑ and R↓ are complementary and the growth ofone is on the expense of the other and happens according to theKAI dependence Equation (3). We can now write the reciprocalvalue of the resistance as a function of the normalized polariza-tion P(t)/Ps = P(t), which represents the time evolution of theswitched domains by an applied pulse:

1/R = 1

R↑+ 1

R↓= 1 − P(t)

RON+ P(t)

ROFF(6)

From Equations (5) and (6) the time dependence of the normal-ized resistance can be written as (see also supplementary onlinematerial):

RN = RONP(t)

ROFF + (RON−ROFF)P(t)= RON

ROFF

P(t)

1 +(

RONROFF

− 1)P(t)

(7)In the case of very large TER, Equation (7) simplifies to:

RN = RON

ROFF

P(t)

1 − P(t)(8)

Equations (7) or (8) together with Equation (3) represent thecharacteristic transfer function of the ferroelectric memristor:

RN = RON

ROFF

(1−e−(t/τ)n

)

1+(

RONROFF

−1) (

1−e−(t/τ)n) ≈ RON

ROFF

(e(t/τ)n+1

)

(9)The experimental data can be fitted with Equation (9) in orderto obtain the chatacteristic value of the switching time τ. As weknow that in ultrathin epitaxial ferroelectric films the polarizationswitching is mostly bi-dimensional, the value of the parameter nin the above relation can be assumed to be 2, and thus one canconstrain the fitting only to one parameter τ. As it can be easily

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Quindeau et al. Programmable ferroelectric tunnel memristor

FIGURE 5 | Memristor programming. The data points represent theresistance after switching from a pre-defined ON-state with a pulse widthof tA = 8 · 10−7 s (red) and tB = 16 · 10−7 s (blue). The vertical linesbetween the data points are the indicators for pulses that were applied.The horizontal lines are the programmed memristor states. The long timerelaxation after applying the switching pulse is related to back-switching andrelaxation of polarization [24].

seen in Figure 3C, the fit of the experimental data is very goodwhich we interpret as the univocal result of ferroelectric domainwall kinetics. The value of the characteristic switching time is inthe range of sub-μs wich is, as expected, similar to the character-istic switching time in thin ferroelectric epitaxial PZT films of thesame composition [23].

The analytical expression (9) can be used to predict or—betterto say—to program any desired memristor state, independentfrom the memristor history. To show this experimentally, we ana-lyzed the response of a ferroelectric memristor by successivelyapplying programming pulses of constant amplitude, startingfrom a well-defined reference state ON, using different pulsewidth tA = 800 ns and tB = 1600 ns. In Figure 5, we present thetime evolution of the resistance of a junction, where it can be eas-ily seen that memristor states marked by horizontal lines can beachieved either with a long pulse tB or with a larger number ofshorter pulses tA demonstrating the ability of programming anyintermediate state between ROFF and RON .

According to the model, applying one programming pulsewith the total width tt should always drive the FTJ to the sameresistance as a sequence of pulses with width tf < tt of which thetotal time is tt = ∑

n tf where n = tt/tf .

CONCLUSIONIn summary, we have shown that a ferroelectric tunnel junctionmemristor based on epitaxial ferroelectric oxide ultra-thin filmscan be programmed in a designed way. The characteristic trans-fer function of this device is directly related to the polarizationswitching mechanism, viz. to the Kolmogorov-Avrami-Ishibashimodel with two-dimensional domain growth. The characteris-tic transfer function of the FTJ memristor is dependent only onthe characteristic switching time of the ferroelectric polarization

which is in the present case of the order of 10−7 s. The phys-ical model of MTJ-based memristors proposed above satisfiesthe mathematical equations describing the general features ofmemristors.

In comparison to other memristor devices, [3] the presentdevice is based on a genuine electronic effect, respectively onquantum electronic transport and does not rely on ionic trans-port or phase change. The writing process, i.e., the partial switch-ing of ferroelectric polarization, is achieved with rather largevoltage approaching the coercive voltage of the ferroelectric layer,but still sufficiently low to comply to the nowadays microelec-tronic requirements. The current-based nondestructive readout,although performed at low voltages (<0.2 V) provides suffi-cient current density to allow downscaling and high integrationdensity. However, further investigations have to be performedin order to identify and understand issues related to reliably,polarization back-switching, fatigue phenomena, etc.

AUTHOR CONTRIBUTIONSAndy Quindeau, PhD student at the Max Planck Instituteof Microstructure Physics. Dietrich Hesse, Professor at theMax Planck Institute of Microstructure Physics. Marin Alexe,Professor at Warwick University.

ACKNOWLEDGMENTSThis research was financed by the German Research Foundation(DFG) via SFB762. The authors acknowledge D. Pantel and S.Götze for the TER sample.

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Conflict of Interest Statement: The authors declare that the research was con-ducted in the absence of any commercial or financial relationships that could beconstrued as a potential conflict of interest.

Received: 23 October 2013; accepted: 03 February 2014; published online: 25February 2014.Citation: Quindeau A, Hesse D and Alexe M (2014) Programmable ferroelectrictunnel memristor. Front. Physics 2:7. doi: 10.3389/fphy.2014.00007This article was submitted to Condensed Matter Physics, a section of the journalFrontiers in Physics.Copyright © 2014 Quindeau, Hesse and Alexe. This is an open-access article dis-tributed under the terms of the Creative Commons Attribution License (CC BY).The use, distribution or reproduction in other forums is permitted, provided theoriginal author(s) or licensor are credited and that the original publication in thisjournal is cited, in accordance with accepted academic practice. No use, distributionor reproduction is permitted which does not comply with these terms.

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